Av(12345, 12435, 13245, 13425, 13452, 14235, 14325, 14352, 41235, 41325, 41352)
Generating Function
\(\displaystyle \frac{-\sqrt{6 x -1}\, \left(2 x -1\right)^{\frac{3}{2}}+6 x^{3}-10 x^{2}+2 x +1}{2 x \left(3 x -2\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 109, 522, 2574, 12964, 66426, 345300, 1816976, 9660732, 51825093, 280168474, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(3 x -2\right)^{2} F \left(x
\right)^{2}+\left(-6 x^{3}+10 x^{2}-2 x -1\right) F \! \left(x \right)+x^{3}-2 x^{2}-x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{18 \left(n + 1\right) a{\left(n \right)}}{n + 4} - \frac{3 \left(8 n + 13\right) a{\left(n + 1 \right)}}{n + 4} + \frac{\left(19 n + 51\right) a{\left(n + 2 \right)}}{2 \left(n + 4\right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{18 \left(n + 1\right) a{\left(n \right)}}{n + 4} - \frac{3 \left(8 n + 13\right) a{\left(n + 1 \right)}}{n + 4} + \frac{\left(19 n + 51\right) a{\left(n + 2 \right)}}{2 \left(n + 4\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 66 rules.
Finding the specification took 2980 seconds.
Copy 66 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{55}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= -\frac{-F_{23}\! \left(x , y\right) y +F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{24}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x , y\right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{38}\! \left(x , y\right) &= -\frac{-F_{39}\! \left(x , y\right) y +F_{39}\! \left(x , 1\right)}{-1+y}\\
F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= -\frac{-y F_{45}\! \left(x , y\right)+F_{45}\! \left(x , 1\right)}{-1+y}\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{52}\! \left(x \right) F_{55}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{52}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= x\\
F_{56}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{55}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{65}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 41 rules.
Finding the specification took 139 seconds.
Copy 41 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{20}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y_{0}\right) &= -\frac{-F_{7}\! \left(x , y_{0}\right) y_{0}+F_{7}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{7}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{8}\! \left(x , y_{0}\right) &= F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right) F_{7}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right)\\
F_{12}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\
F_{13}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}\right) F_{20}\! \left(x \right)\\
F_{15}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{0}\right)+F_{18}\! \left(x , y_{0}\right)+F_{21}\! \left(x , y_{0}\right)\\
F_{16}\! \left(x , y_{0}\right) &= F_{17}\! \left(x , y_{0}\right)\\
F_{17}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\
F_{18}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right) F_{20}\! \left(x \right)\\
F_{19}\! \left(x , y_{0}\right) &= -\frac{-F_{6}\! \left(x , y_{0}\right) y_{0}+F_{6}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{22}\! \left(x , y_{0}\right)\\
F_{22}\! \left(x , y_{0}\right) &= F_{23}\! \left(x , y_{0}, 1\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{24}\! \left(x , y_{0}, y_{1}\right) y_{1}+F_{24}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{0}, y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{7}\! \left(x , y_{0}\right) y_{0}-y_{1} F_{7}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{1}\right) F_{29}\! \left(x , y_{0}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{1}, y_{0}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{27}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{27}\! \left(x , y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x , 1\right)\\
F_{35}\! \left(x , y_{0}\right) &= -\frac{-F_{36}\! \left(x , y_{0}\right) y_{0}+F_{36}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{36}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x , y_{0}\right)+F_{40}\! \left(x , y_{0}\right)\\
F_{37}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right) F_{38}\! \left(x , y_{0}\right)\\
F_{38}\! \left(x , y_{0}\right) &= F_{39}\! \left(x , 1, y_{0}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{7}\! \left(x , y_{0} y_{1}\right) y_{0}+F_{7}\! \left(x , y_{1}\right)}{-1+y_{0}}\\
F_{40}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{35}\! \left(x , y_{0}\right)\\
\end{align*}\)